toss 5 fair coins and let X equal the number of heads observed.

Question

A. Identify the  sample points associated with this experiment and assign a value of x to each sample Point.

B. calculate P (X) for the values x =1 & x = 3

D. What is P(x=3 or x = 4)?

Answer 1

The sample points would be all combinations of Heads or Tails for 5 coins, so there are 2⁵ possible outcome for the set of heads and tails, { (HHHHH), (HHHHT)...(TTTTT)}

 

For this question, X is the number of heads, which means there could be

0 heads : TTTTT     this accounts for one of the 32 possible outcomes of 2⁵

5 heads : HHHHH   this accounts for one of the 32 possible outcomes of 2⁵

1 Head  : the one head could occur in any one of the 5 slots, so this is 5 of the 32 outcomes

4 Heads : the one tail could occur in any of the 5 slots, so this is 5 of the 32 outcomes

 

This leaves 2 heads (and three tails) and 3 heads (or 2 tails). Not a couple of things here,

the number of ways to select 2 heads has to be the same number of ways to select 3 tails; and

the number of ways to select 3 heads has to be the same number of ways to select 2 tails,

so the P(2 heads ) has to equal P(3 heads).

 Out of the 32 possible combinations we know 1+1+5+5 = 12, so each of the remaining is (32-12)/2 = 10

P(2)=p(3) = 10.

 

But check anyway: to choose 3 out of 5 = 5*4*3/(3*2*1) = 10

to choose 2 out of 5 = 5*4/(2*1) = 10